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The continuum reaction-diffusion limit of a stochastic cellular growth model

Stephan Luckhaus, Livio Triolo (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A competition-diffusion system, where populations of healthy and malignant cells compete and move on a neutral matrix, is analyzed. A coupled system of degenerate nonlinear parabolic equations is derived through a scaling procedure from the microscopic, Markovian dynamics. The healthy cells move much slower than the malignant ones, such that no diffusion for their density survives in the limit. The malignant cells may locally accumulate, while for the healthy ones an exclusion rule is considered....

The dynamics of weakly interacting fronts in an adsorbate-induced phase transition model

Shin-Ichiro Ei, Tohru Tsujikawa (2009)

Kybernetika

Hildebrand et al. (1999) proposed an adsorbate-induced phase transition model. For this model, Takei et al. (2005) found several stationary and evolutionary patterns by numerical simulations. Due to bistability of the system, there appears a phase separation phenomenon and an interface separating these phases. In this paper, we introduce the equation describing the motion of two interfaces in $ℝ^{2}$ and discuss an application. Moreover, we prove the existence of the traveling front solution which approximates...

The Feynman-Kac formula for a system of parabolic equations

Charles S. Kahane (1994)

Czechoslovak Mathematical Journal

The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi, Giuseppe Mingione (2014)

Journal of the European Mathematical Society

The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of $p$ -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

Time-dependent invariant regions for parabolic systems related to one- dimensional nonlinear elasticity

Eduard Feireisl (1990)

Aplikace matematiky

A parabolic system arisng as a viscosity regularization of the quasilinear one-dimensional telegraph equation is considered. The existence of $L \infty$ - a priori estimates, independent of viscosity, is shown. The results are achieved by means of generalized invariant regions.